Considerable advances have been made in the past two decades in developing advanced techniques for numerical simulation of fluid flows in aerodynamic configurations. These techniques are now mature enough to be used routinely, in conjunction with experimental results, in aerodynamic design. However, aerodynamic design optimization procedures that make efficient use of these advanced techniques are still being developed.
The design of aircraft components, such as a wing, a fuselage or an engine, involves obtaining an optimal component shape that can deliver the desired level of component performance, subject to one or more constraints (such as maximum weight or cost) that the component(s) must satisfy. Aerodynamic design can be formulated as an optimization problem that requires minimization of an objective function, subject to constraints. Many formal optimization methods have been developed and applied to aerodynamic design. These include inverse design methods, adjoint methods, sensitivity derivative-based methods and traditional response surface methodology (RSM).
Inverse design methods in aerodynamics are used to provide a component that responds in a pre-selected manner, for example, an aircraft wing that has a prescribed pressure distribution. The known inverse methods do not account for certain fluid parameters, such as viscosity, and are used in preliminary design only.
Adjoint methods provide a designer with the gradient of the objective function. One advantage of this method is that the gradient information is obtained very quickly. However, where several technical disciplines are applied simultaneously, it is often difficult to perform design optimization using this method; each discipline requires a different formulation. It is also difficult and expensive to quickly evaluate the effects of engineering tradeoffs, where the applicable constraints may be changed several times. It is also not possible to use existing experimental data or partial or unstructured data in the design process.
A sensitivity derivative-based method typically requires that a multiplicity of solutions, with one parameter varied at a time, be obtained to compute a gradient of the objective function. The number of computations required grows linearly with the number of design parameters considered for optimization, and this method quickly becomes computationally expensive. This method is also sensitive to noise present in the design data sets. As with an adjoint method, it is not possible to use existing experimental data or partial or unstructured data in the design process.
RSM provides a framework for obtaining an optimal design, using statistical procedures, such as regression analysis and design of experiments. Traditional RSM uses low-degree regression polynomials in the relevant design variables to model the variation of an objective function. The polynomial model is then analyzed to obtain an optimal design. Several polynomial models may have to be constructed to provide an adequate view of the design space. Addition of higher degree polynomials will increase the computational cost and will build in higher sensitivity to noise in the data used.
Artificial neural networks (“neural nets” herein) have been widely used in fields such as aerodynamic engineering, for modeling and analysis of flow control, estimation of aerodynamic coefficients, grid generation and data interpolation. Neural nets have been used in RSM-based design optimization, to replace or complement a polynomial-based regression analysis. Current applications of neural nets are limited to simple designs involving only a few design parameters. The number of data sets required for adequate modeling may increase geometrically or exponentially with the number of design parameters examined. A neural net analysis requires that the design space be populated with sufficiently dense simulation and/or experimental data. Use of sparse data may result in an inaccurate representation of the objective function in design space. On the other hand, inefficient use of design data in populating the design space can result in excessive simulation costs. Capacity control is critical to obtain good generalization capability. In some preceding work, this problem was alleviated by using a neural net to represent the functional behavior with respect to only those variables that result in complex, as opposed to simple, variations of the objective function; the functional behavior of the remaining variables was modeled using low degree polynomials. This requires a priori knowledge to partition the design variables into two sets.
FIG. 1 graphically illustrates results of applying a simple NN analysis to a one-parameter model, namely, an approximation to the second degree polynomial y=2·(0.5−x)2 at each of 3 pairs of training values (curve A) and at each of 5 pairs of training values (curve B). Use of more than the minimum number (3) of training pairs clearly improves the fit over the domain of the variable x. It is theoretically possible that only Q+1 spaced apart training value pairs are needed to completely specify a Qth degree polynomial (for example, Q=6). However, because of the presence of noise, the theoretical minimum number of training value pairs is seldom sufficient to provide an acceptable fit.
Use of neural network (NN) analysis of a physical object, in order to optimize response of the object in a specified physical environment, is well known. An example is optimization of a turbine blade shape, in two or three dimensions, in order to reproduce an idealized pressure distribution along the blade surface, as disclosed by Rai and Madavan in “Aerodynamic Design Using Neural Networks”, AIAA Jour., vol. 38 (2000) pp. 173-182. NN analysis is suitable for multidimensional interpolation of data that lack structure and provides a natural structure in which a succession of numerical solutions of increasing complexity, or increasing fidelity to a real world environment, can be represented and optimized. NN analysis is especially useful when multiple design objectives need to be met.
A feed-forward neural net is a nonlinear estimation technique. One difficulty associated with use of a feed-forward neural net arises from the need for nonlinear optimization to determine connection weights between input, intermediate and output variables. The training process can be very expensive when large amounts of data need to be modeled.
In response to this, a support vector machine (SVM) approach, originally applied in statistical learning theory, has been developed and applied. Support vector machine analysis allows use of a feature space with a large dimension, through use of a mapping from input space into feature space and use of a dual formulation of the governing equations and constraints. One advantage of an SVM approach is that the objective function (which is to be minimized to obtain the coefficients that define the SVM model) is convex so that any local minimum is also a global minimum; this is not true for many neural net models. However, an underlying feature space (polynomial, Gaussian, etc.) must be specified in a conventional SVM approach, and data resampling is required to implement model hybridization. Hybridization is more naturally, and less expensively, applied in a neural net analysis.
What is needed is a machine learning algorithm that combines the desirable features of NN analysis and of SVM analysis and does not require intimate a priori familiarity with operational details of the object to be optimized. Preferably, the method should automatically provide a characterization of many or all of the aspects in feature space needed for the analysis.